Take in and with . Then and by 2.1.3 Lemma (3) and its proof for some self-adjoint element . Hence This shows that is open relative to .
The complement is a disjoint union of cosets of the form with . Each of these cosets are homeomorphic to and therefore open relative to , consequently is closed in .
In conclusion, is a non-empty subse of , is closed and open in , and is connected. This implies that .
Therefore 2. and 3. hold.